Download A single molecule rectifier with strong push-pull coupling

THE JOURNAL OF CHEMICAL PHYSICS 129, 204701 共2008兲
A single molecule rectifier with strong push-pull coupling
Aldilene Saraiva-Souza,1 Fabricio Macedo de Souza,2 Vicente F. P. Aleixo,3
Eduardo Costa Girão,1 Josué Mendes Filho,1 Vincent Meunier,4 Bobby G. Sumpter,4
Antônio Gomes Souza Filho,1,a兲 and Jordan Del Nero5,a兲
1
Departamento de Física, Universidade Federal do Ceará, Fortaleza, Ceará 60455-900, Brazil
Centro Internacional de Física da Matéria Condensada, Universidade de Brasília,
70904-910 Brasília, Brazil
3
Pós-graduação em Engenharia Elétrica, Universidade Federal do Pará, Belém, Pará 66075-900, Brazil
4
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
5
Departamento de Física, Universidade Federal do Pará, Belém, Pará 66075-110, Brazil
2
共Received 22 August 2008; accepted 17 October 2008; published online 24 November 2008兲
We theoretically investigate the electronic charge transport in a molecular system composed of a
donor group 共dinitrobenzene兲 coupled to an acceptor group 共dihydrophenazine兲 via a polyenic chain
共unsaturated carbon bridge兲. Ab initio calculations based on the Hartree–Fock approximations are
performed to investigate the distribution of electron states over the molecule in the presence of an
external electric field. For small bridge lengths 共n = 0 – 3兲 we find a homogeneous distribution of the
frontier molecular orbitals, while for n ⬎ 3 a strong localization of the lowest unoccupied molecular
orbital is found. The localized orbitals in between the donor and acceptor groups act as conduction
channels when an external electric field is applied. We also calculate the rectification behavior of
this system by evaluating the charge accumulated in the donor and acceptor groups as a function of
the external electric field. Finally, we propose a phenomenological model based on nonequilibrium
Green’s function to rationalize the ab initio findings. © 2008 American Institute of Physics.
关DOI: 10.1063/1.3020353兴
I. INTRODUCTION
Electronic transport through single molecules is currently a subject of intense study mainly due to its potential
application for new electronic devices such as molecular
rectifiers.1–4 The ability of molecules to develop strong rectification when attached to donor and acceptor groups was
first proposed in the seminal work of Aviran and Ratner.5
Other proposals of molecular rectifiers, diodes, and fieldeffect transistors have also been reported in the literature.6 In
particular, push-pull with a ␴ or ␲ bridge have been studied
with special attention on the effects of the bridge length on
the transport properties.7 Recently, it was pointed out that
systems with donor and acceptor groups spaced by a carbon
␲ bridge have strong rectification for relatively long bridges,
thus operating as molecular field-effect transistors.8 In addition it was observed that ␲ bridges are more favorable to
rectification than ␴ bridges.9,10 In contrast to the results presented in Ref. 11, here we show that the system
dinitrobenzene共D兲-bridge-dihydrophenazine共A兲 has a strong
current rectification even for small ␲ bridges, thus resulting
in a very robust rectification compared to similar proposals
reported in the literature.
We focus our analysis on the frontier molecular orbitals
共FMOs兲 关highest occupied molecular orbital 共HOMO兲 and
lowest unoccupied molecular orbital 共LUMO兲, in particular兴
and the charge distribution in the molecule, since they can
provide complementary information regarding electronic
a兲
Authors to whom correspondence should be addressed. Electronic addresses: [email protected] and [email protected].
0021-9606/2008/129共20兲/204701/6/$23.00
transport in the molecule.11 It is important to mention that an
analysis of the HOMO/LUMO frontier orbitals does not always provide the actual tendency of the electron flow. For
instance, in the present system for n = 0 and n = 1, we find a
uniform distribution of the FMOs with high level energies
共typically −1 eV兲, which in a first analysis would predict no
effective flow of electrons in the molecule and a uniform
charge distribution. In fact, we will see that a strong asymmetry in the charge distribution is observed in the system
studied here even for a small bridge length. By increasing the
bridge length between the donor and acceptor, a localized
HOMO and LUMO conduction channel becomes evident in
the charge versus voltage curves, thus leading to sharp steps
typically observed in resonant level models.
In order to understand electronic transport behavior of
the system, we have developed a formulation based on nonequilibrium Green’s function that describes transport through
localized levels coupled to a donor and to an acceptor of
electrons. This theoretical approach allowed us to gain further insight and a clear physical interpretation of the rectification effects obtained from ab initio calculations.
The paper is organized as follows. In Sec. II we introduce the adopted methodology with a brief description of the
computational package used and the formulation based on
the nonequilibrium Green’s function. In Sec. III we present
our results and the corresponding discussion, and in Sec. IV
we make the final remarks.
II. METHODOLOGY
Two theoretical approaches are adopted in our study: 共i兲
ab initio calculations and 共ii兲 nonequilibrium Green’s func-
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204701-2
J. Chem. Phys. 129, 204701 共2008兲
Saraiva-Souza et al.
tion. In the first one we use the GAUSSIAN98 package,12 which
encompasses Hartree–Fock 共HF兲 in its numerical procedure.
We have used the HF/6-31G and HF/ 6-31Gⴱ and the results
obtained are in agreement so we adopted the HF/6-31G,
which has been successfully applied for determining electronic structure in these types of systems.13 We have used HF
instead of density functional theory because hybrid
exchange-correlation functionals, local density approximation, and generalized gradient approximation, are well
known to overestimate the polarizabilities of quasilinear systems and do not reproduce the localization of the molecular
orbitals and inversion behavior observed with HF-based
methods.14–17 The effects of an external electric field and its
interplay with charging accumulation in the donor/acceptor
group are considered in part 共i兲. Additionally, we determine
the charge accumulated 共Q兲 in the acceptor group as a function of the external electric field 共Q-V curve兲. In the second
part 关共ii兲 above兴 we develop a simple model consisting of
multiple electronic levels coupled to a donor and to an acceptor group. By adjusting phenomenological coupling parameters, we fit the equilibrium Q-V curve obtained in 共i兲
with a nonequilibrium I-V curve determined via Green’s
functions. Surprisingly, both HF and transport calculations
lead to similar results; therefore, this agreement reveals that
charge localization can provide valuable information about
transport properties of the present system. However, a general extension of this equivalence to other systems is not
straightforward.18
The computational approach of connecting ab initio results with nonlinear Green’s function has been successfully
used in previous works.19,20 Our transport model is based on
a ballistic resonant tunneling transport from a donor to an
acceptor of electrons through localized levels weakly or
strongly coupled to the reservoirs 关see Fig. 1共a兲兴. This approach can be done by ignoring molecular charging effects
because of the fast electronic switching on the molecular
structure and because the transport is only associated with
the molecular electronic structure.21 Figure 1共b兲 shows schematically the nonequilibrium system used to generate an I-V
curve that resembles the equilibrium Q-V curve. Three levels
共E1 , E2 , E3兲 of different widths are coupled to a left and to a
right lead via tunneling barriers. The left and right Fermi
energies are denoted by EFL and EFR, respectively. These energies are related via EFL − EFR = eV, where V is the left-right
共bias兲 voltage and e the electron charge 共e ⬎ 0兲. Deep levels
are narrow while the level closer to the top of the tunneling
barriers is broad. The narrow levels give rise to sharp steps in
the characteristic I-V curve whenever these levels match the
emitter chemical potential 共the donor for positive and the
acceptor for negative bias兲. On the other hand, the broad
level contributes to a linear increase in the current even
though it remains above the emitter Fermi energy for the bias
voltages used. Therefore this broad level is responsible
for the nonzero slope of the plateaus in the I-V curves discussed below.
FIG. 1. 共Color online兲 共a兲 Sketch of the resonant tunneling system used to
model the donor-bridge-acceptor system shown in 共b兲. Three levels are
coupled to a left 共l兲 and to a right 共r兲 electron reservoir via asymmetric
tunneling barriers. In the presence of a bias voltage the left and the right
Fermi energies, ELF and ERF, respectively, differ from each other, thus generating a current when resonances are matched.
The system is described by the following Hamiltonian:
H = HL + HR + H M + Hhib ,
where HL, HR, and H M are the Hamiltonians of the donor, the
acceptor, and the molecule, respectively. The term Hhib accounts for the hybridization of molecular orbitals to the
donor/acceptor orbitals. This term gives rise to a charge current in the presence of bias voltages. Indeed, these terms read
HL/R = 兺 E␣c␣+ L/Rc␣L/R ,
␣
H M = 兺 End+n dn ,
Hhib = 兺 关tc␣+ Ldn + tⴱd+n c␣L + tc␣+ Rdn + tⴱd+n c␣R兴,
␣n
where E␣ and En are the leads and the molecular levels,
respectively. To calculate the electronic current in this system, we use the following equation:
IL/R = − e具dNL/R/dt典 = − ie具关NL/R,H兴典,
where NL/R is the total number of operators for electrons in
the donor 共L兲 or acceptor 共R兲. The notation 具¯典 denotes
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204701-3
J. Chem. Phys. 129, 204701 共2008兲
Single molecule rectifier with strong push-pull coupling
some nonequilibrium thermodynamic average and 关¯兴 denotes a commutator. Following the standard procedure of
nonequilibrium Green’s functions,22,23 the current is given by
IL/R = 2e
冕
⌫␩ =
dET共E兲关f L/R共E兲 − f R/L共E兲兴,
1
.
1 + exp关共E − ␮L/R兲兴/关kBT兴
冤
␩
⌫2
0
¯
0
¯
␩
0
0
⌫3 ¯
]
]
]
冥
,
GrM = 关EI − HM − ⌺r兴−1 ,
␮L and ␮R denote the chemical potentials and kB the Boltzmann constant. The parameters ⌫L and ⌫R give the tunneling
rates between left/right and the molecule 共bridge兲. In a matrix notation they are given by
GrM =
0
0
where ␩ ⬅ L or R. The molecular retarded Green’s function
GrM is given by
where T共E兲 = Tr兵⌫LGrM ⌫RGaM 其 and f L and f R are the left and
right Fermi distribution functions, respectively, given by
f L/R共E兲 =
冤
⌫1␩
E − E1 + i⌫/2
0
0
¯
0
E − E2 + i⌫/2
0
¯
0
0
]
]
E − E3 + i⌫/2 ¯
]
冥
This function is appropriate for describing many noninteracting levels weakly coupled to reservoirs. The advanced
Green’s function is the complex conjugate of GrM , i.e., GaM
ⴱ
= GrM . We assume a linear bias-voltage 共eV兲 drop along the
system. So ␮L = ␮L0 and ␮R = ␮R0 − eV, where ␮L0 and ␮R0 are
the left and right chemical potentials. The levels En are given
by En = E0n − x eV, where x accounts for bias-voltage drop
asymmetry. Asymmetries in the voltage drop can happen, for
instance, when the molecular levels are more coupled to the
donor than to the acceptor or the opposite. In particular, for a
symmetric drop we have x = 0.5. It is important to mention
that molecular rectifiers based on asymmetries of the leadsto-molecule coupling have been predicted, for instance, in a
carbon conjugated-bond bridge24 and in the “tour wire.”25
The main role of the asymmetry of the potential drop relies
on the fact that the molecular levels can line up either with
the left- or the right-lead Fermi level differently depending
on the sign of potential energy eV.
The parameters in the transport model, used to fit the
numerical ab initio results, are En, x, and ⌫. Basically, En and
x give the resonance positions on the I-V characteristic
curve, while ⌫ provides the intensity of the current.
III. RESULTS AND DISCUSSION
Figure 2 shows the HOMO and LUMO energies for different bridge lengths 共n = 0 , 1 , . . . , 10兲. As the bridge increases the HOMO 共LUMO兲 energy increases 共decreases兲
asymptotically to a value slightly above 8.0共−1.7兲 eV. This
is a general behavior of quantum confined systems that has
its energies reduced whenever the confinement length increases. This suppression leads to a more efficient electron
where I is the identity matrix and ⌺r = −i共⌫L + ⌫R兲 / 2 =
−i⌫ / 2 is the tunneling self-energy. This quantity is assumed
to be energy independent 共wideband limit兲. In a matrix notation this Green’s function can be written as
−1
.
transport reaching a limiting value. In Fig. 2 共inside panels兲
we plot the FMOs for n = 0 共I兲, n = 1 共II兲, and n = 6 共III兲. This
is a relevant quantity to consider since it provides information about the charge distribution along the molecule for the
molecular ground state. We note that when the bridge increases, the orbitals tend to localize on the acceptor 共donor兲
side for LUMO 共HOMO兲. This indicates that in the presence
of an electric field, the charge will be predominantly accumulated on one side of the system, as is shown below.
In Fig. 3共a兲 we show the charge accumulated 共Q兲 in the
acceptor as a function of external electric field 共Q-V curve兲.
For positive electric field the charge is accumulated in the A
group and depleted in group D, thus generating a polarization in the molecule. For negative fields the electron flow is
reversed with depletion in the acceptor side. Interestingly,
there is an asymmetry from positive to negative fields, which
becomes more pronounced for n = 2 and 3. This asymmetry
in the charge response is closely related to the orbitals illustrated in Fig. 2. Within the FMOs, localization allows the
identification of the electronic flow direction because molecular structures with FMO delocalized over the structure
indicate no particular flow direction. Therefore this system
can potentially operate as a single bidirectional molecular
field-effect transistor.
We note that for n = 2 and 3 the electrons in D group do
not respond to relatively small positive fields. The onset of
electron transport from D to A starts around 4 V/nm and then
becomes much stronger than its negative counterpart, thus
revealing a considerable rectification. The strength of the
charge accumulation is larger for positive bias since we are
imposing a flow in a direction, which has more FMOs avail-
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204701-4
Saraiva-Souza et al.
FIG. 2. 共Color online兲 共Left兲 HOMO 共top panel兲 and LUMO 共bottom panel兲
energies for the molecular structures with bridges n = 0 , . . . , 10 and 共right兲
charge distribution of FMO for n = 0 共I兲, n = 1 共II兲, and n = 6 共III兲.
able to populate. This simple picture, though, does not provide a qualitative explanation for the resonances in the Q-V
curve seen at 4 V/nm and around 14 V/nm for n = 3. These
resonance matches in the Q-V curve become more evident
when the bridge length is increased.
Figures 3共b兲 and 3共c兲 show Q-V curves for bridge
lengths ranging from 4 to 10. The Q-V curves reveal resonances and plateaus, which is typical for systems with levels
coupled to donor and to acceptor leads.26 In addition to the
Q-V curves, also we present the I-V characteristic curves for
two selected bridge values, n = 4 and 10 共Fig. 4兲. These results reveal that an equilibrium quantity such as the charge
accumulation Q can provide, in this case, information about
nonequilibrium quantities such as I. The current was calculated here via Green’s formulation previously presented and
applied to a model system illustrated in Fig. 1. The I-V
curves were generated considering levels of different widths
coupled to both donor and acceptor via asymmetric tunneling
barriers. Differences in the widths can be understood by noting that higher levels are closer to the top of the tunnel barriers. The corresponding wave functions tend to overlap
more strongly with the donor and acceptor electron wave
J. Chem. Phys. 129, 204701 共2008兲
FIG. 3. 共Color online兲 共a兲 In n = 0 and n = 1, negative external electric field
D ← A 共E−兲 when the A group is in a potential of reverse polarization.
Positive external electric field D → A 共E+兲 when the A group is in a potential
of forward polarization. For the same applied potential the charge accumulation is greater than the negative one exhibiting a typical rectifier signature.
In n = 2 and n = 3 prominent rectifications of E− and E+ with resonance
region;
共b兲
negative
external
electric
field
in
the
E+
D ← A 共E−兲 when the A group is in a potential of reverse polarization.
Positive external electric field D → A 共E+兲 when the A group is in a potential
of forward polarization followed by resonances with intense saturation values. 共c兲 Decrease in the rectification potential generated with a weak coupling because of the noninteracting DA groups.
function, which is equivalent to a stronger coupling and consequently a broadening of those levels 共longer lifetime兲. On
the other hand, deep levels are more localized, thus resulting
in narrower levels. While the broad levels give rise to the
nonzero slope of the plateaus, the narrow ones result in the
sharp steps observed in Fig. 4. The numerical fitting parameters adjusted to best fit the I-V with the Q-V curves are the
energy levels E0n, the coupling parameters ⌫L/R
n , and x. Their
values are listed in Table I.
Related studies have been made by Mujica et al.27 where
current versus voltage curves, including tunneling and Coulomb repulsion, were examined using a Hubbard model
treated at the HF level. The results showed negative differential resistance associated with increased localization of the
molecular resonances. Zahid et al.28 found a similar effect by
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J. Chem. Phys. 129, 204701 共2008兲
Single molecule rectifier with strong push-pull coupling
FIG. 4. 共Color online兲 Two selected curves comparing the charge distribution 共dots兲 in the acceptor group for 共a兲 n = 4 and 共b兲 n = 10 and the current
共solid full line兲.
as conduction channels when an external electric field is applied. The unusual charge transfer of these molecules under
an external electric field is remarkably different from those
of conjugated compounds of similar size. Also, the dependence of the D and A groups from their traditional role of
electron flow opens up the possibility of exploiting very interesting molecular devices involving a new manifold of excited states now accessible in an unusual manner.
It is important to point out a recent study30 where the
unusual I-V characteristics of acyclic cross-conjugated molecules were found to be caused by quantum interference. In
our work we find that there are two different electrical behaviors for small and large bridges. The nonequilibrium/
equilibrium characteristics reveal an induced charge transfer
highly dependent on molecular resonances.
ACKNOWLEDGMENTS
using a master equation applied to a prototypical benzene
molecule and further showed a strong correlation between
the molecule-electrode coupling. Also, a theoretical/
experimental study of symmetric molecules has been done,29
showing the same qualitative results as we describe here, i.e.,
共i兲 small rectification behavior for forward and reverse bias
and 共ii兲 the same microampere current level at small
voltages.
IV. CONCLUSIONS
We have reported ab initio results for a single molecule
system that exhibits rectification due to strong push-pull coupling. For small bridge lengths 共n = 0 – 3兲 we found a homogeneous distribution of the FMO, while for n ⬎ 3 there was a
strong localization of the LUMO orbitals. Interestingly, the
localized orbitals between the donor and acceptor groups act
TABLE I. Numerical parameters used in the transport model to better fit the
ab initio calculations.
Physical meaning
Tunneling rates between the levels
and the donor/acceptor
Quantity/values
⌫0 = 2.15 meV
⌫D1 = 0.8⌫0, ⌫A1 = 0.8⌫0
⌫D2 = ⌫0, ⌫A2 = ⌫0
⌫D3 = 170⌫0, ⌫A3 = ⌫0
⌫0 = 2.15 meV
⌫D1 = ⌫0, ⌫A1 = 0.8⌫0
⌫D2 = ⌫0, ⌫A2 = ⌫0
⌫D3 = 150⌫0, ⌫A3 = 150⌫0
⌫0 = 2.15 meV
⌫D1 = ⌫0, ⌫A1 = 0.8⌫0
⌫D2 = 2⌫0, ⌫A2 = ⌫0
⌫D3 = 150⌫0, ⌫A3 = 150⌫0
n
4
7
10
Level energies
E01 = 2.65, E02 = 7.81, E03 = 15 eV
E01 = 1.98, E02 = 4.35, E03 = 11 eV
E01 = 1.54, E02 = 3.4, E03 = 11 eV
4
7
10
Potential asymmetry parameter
x = 0.68
x = 0.6
x = 0.6
4
7
10
A.S.-S. is grateful to a CNPq fellowship. F.M.S. acknowledges IBEM agency. A.G.S.F. and J.M.F. acknowledge
the FUNCAP and CNPq agencies. J.D.N. and V.F.P.A. acknowledge the FAPESPA agency. A.G.S.F., J.M.F., and
J.D.N. acknowledge the Rede Nanotubos de Carbono/CNPq.
This research was supported in part by the Division of Materials Science and Engineering, U.S. Department of Energy
and the Center for Nanophase Materials Sciences 共CNMS兲,
sponsored by the Division of Scientific User Facilities, U.S.
Department of Energy under Contract No. DEAC05–
00OR22725 with UT-Battelle, LLC at Oak Ridge National
Laboratory.
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